Slide Rule Multiply

A slide rule on a bench stand, showing how sliding one logarithmic scale along another adds two lengths, and adding those lengths multiplies the two numbers.

Hairline: lands on D at log a + log b, where the product is readSlide: moved by log a, carrying the C scale against DA slide rule multiplies by adding lengths: log(a × b) = log a + log b. Setting the slide's 1 over a on D and moving the hairline to b on C puts it over a × b on D. When the sum runs past the end of D the slide is set from its right index instead, and the reading must be scaled by ten: the rule only adds lengths and never tracks the decimal point, and an ordinary rule gives about two to three significant figures, so the readout writes D to three figures near 1 and two near 10, adding the true product only where the two differ. The decade length, graduation pattern and reading resolution are illustrative; the glass cursor is a bench-model reading aid, and the faint A and B scales are not used here.

What you're looking at

This is a slide rule, a ruler-like calculator used from the 1620s to multiply and divide quickly by eye. It lies across the bench in two cradles on short posts, running from the back left to the front right, with a card of logarithm tables on the bench to its left and, at the front right, a scratch slip held by a brass weight with a quill. The rule has three strips: the fixed strip nearest you carries the D scale, the centre strip (the slide) carries the C scale and moves along the rule, and the faint A and B scales are not used here. A glass cursor with a thin hairline rides across the rule; you drag either the slide or the cursor, or use the arrow keys, and both parts glide to their new places.

What it means

The marks on C and D are spaced by logarithm, so the distance from 1 to a number stands for its log. Because log(a × b) = log a + log b, moving the slide by the length of a and then the cursor a further length of b puts the hairline at a × b on D: the rule turns multiplication into adding lengths. The guide says an ordinary rule gives only about two to three significant figures and does not track the decimal place, which you must work out yourself. Logarithms date from 1614 (Napier) and the slide rule from the 1620s, and for anyone doing long chains of multiplication by hand they save a great deal of work.

Try this

  1. Drag the slide (the centre strip) along the rule, or press Left and Right (Shift for whole steps) to change a. The slide moves so its 1 sits over a on D, the hairline follows to keep b, and the readout briefly says 'setting the slide' before giving the new reading on D. Moving the slide lays down the first length, log a, the first half of the sum.
  2. Drag the glass cursor along the rule, or press Up and Down (Shift for whole steps) to change b. The hairline moves over b on the C scale, the readout says 'moving the hairline', then shows what D reads under it; at the starting setting, 2.00 × 4.00 gives 'D reads ≈ 8.0'. The cursor adds the second length, log b, so the hairline lands on D at log a + log b, which is the product.
  3. Drag the slide and cursor until a × b passes 10, or press End to set 10 × 10. The slide is now set from its right-hand 10 instead of its 1, and the readout adds 'right index' and gives D's reading with an arrow to ten times that value; at 10 × 10 it reads '10.0 × 10.0 · right index · ≈ 10.0 → 100'. The rule carries only one run of 1 to 10 and cannot show where the decimal point goes, so the user has to shift it.
  4. Leave the figure alone until its tour reaches 1.60 × 4.60, or set it yourself; press Home to go back to 1 × 1. The readout gives 'D reads ≈ 7.4 · true 7.36', because near the top of the scale the rule can be read only to about two figures. This is the guide's limit of about two to three significant figures for an ordinary rule.

Reading the readout

The line starts with the two numbers being multiplied, a × b, each to three figures; while the parts are still moving it adds 'setting the slide' or 'moving the hairline'. Once the hairline rests and the product is under 10, 'D reads ≈' gives the number under the hairline on the D scale, to three figures near 1 and two near 10, because the marks crowd together toward 10 (the reading precision is this drawing's illustrative choice). If the product passes 10, that phrase is replaced by 'right index · ≈', meaning the slide was set from its right-hand 10, and the arrow shows the reading scaled up by ten. In either form, 'true' gives the real product rounded to three figures and appears only when it differs from what the rule shows.

What the model uses

  • log(a × b) = log(a) + log(b): a table of logarithms converts every multiplication into an addition and every division into a subtraction.
  • Logarithms: Napier, 1614. The slide rule: 1620s.
  • The slide rule is two logarithmically graduated sliding scales, so setting one against the other performs the addition mechanically; its usable precision is often around two to three significant figures for an ordinary rule.
  • A slide rule does not track decimal place or validate the model.
  • Illustrative, not from the guide: one decade drawn 250 units long, graduations every 0.1 (1–2), 0.2 (2–5) and 0.5 (5–10), and a reading resolution of about 0.5 units, which sets how finely the readout rounds.

The figure draws one ordinary rule with an illustrative scale length and reading resolution, and simulates only multiplication with the C and D scales; a real rule's precision also depends on graduation, alignment, eyesight and technique, and its result still needs an independent check of the decimal place and of the model it feeds. This is a code-drawn explanatory model for orientation, not an equipment drawing or a safety procedure; follow the guide's warnings.